Question: Simplify and expand the following expression: $ \dfrac{4x - 10}{x - 1}+\dfrac{x + 2}{5x - 5} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(x - 1)(5x - 5)$ Multiply the first term by $\dfrac{5x - 5}{5x - 5}$ $ \begin{align*} \dfrac{4x - 10}{x - 1} \times \dfrac{5x - 5}{5x - 5} & = \dfrac{(4x - 10)(5x - 5)}{(x - 1)(5x - 5)} \\ & = \dfrac{20x^2 - 70x + 50}{(x - 1)(5x - 5)}\end{align*} $ Multiply the second term by $\dfrac{x - 1}{x - 1}$ $ \begin{align*} \dfrac{x + 2}{5x - 5} \times \dfrac{x - 1}{x - 1} & = \dfrac{(x + 2)(x - 1)}{(5x - 5)(x - 1)} \\ & = \dfrac{x^2 + x - 2}{(5x - 5)(x - 1)}\end{align*} $ Now we have: $ = \dfrac{20x^2 - 70x + 50}{(x - 1)(5x - 5)} + \dfrac{x^2 + x - 2}{(5x - 5)(x - 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{20x^2 - 70x + 50 + x^2 + x - 2}{(x - 1)(5x - 5)} $ $ = \dfrac{21x^2 - 69x + 48}{(x - 1)(5x - 5)}$ Expand the denominator: $ = \dfrac{21x^2 - 69x + 48}{5x^2 - 10x + 5}$